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1.) Introduction

We present Motivation for the developments here. Then we we give a leisurely survey of the general abstract theory and one of the concrete implementation in Smooth∞Grpd.

Motivation

There are two different but related motivations for the consideration of differential cohomology in an $\infty$-topos: from gauge theory in fundamental theoretical physics, and from Chern-Weil theory in pure mathematics.

From gauge theory

The development of differential cohomology has and still is to a considerable extent motivated by structures appearing in fundamental physics in general and gauge theory in particular.

• . Higher category theory and physics – Gauge theory

From Chern-Weil theory

Classical Chern-Weil theory and the theory of secondary characteristic classes is a cornerstone of differential geometry. There are however various operations that suggests themselves naturally but are not possible in classical Chern-Weil theory. The theory that we present may be regarded as providing a context more general than classical differential geometry in which these constructions do exist.

This is discussed at

• . ∞-Chern-Weil theory – Motivation

General abstract theory

The framework of all our constructions is topos theory (Elephant) or rather, more generally, (∞,1)-topos theory (HTT)

In the sections Toposes and (∞,1)-Toposes below we leisurely recall and survey basic notions with an eye towards our central example of an (∞,1)-topos: that of smooth ∞-groupoid.

In these sections tthe reader is assumed to be familiar with basic notions of category theory (such as adjoint functors) and basic notions of homotopy theory (such as weak homotopy equivalences). A brief introduction to relevant basic concepts (such as Kan complexes and homotopy pullbacks) is given in section Concrete implementation in Smooth∞Grpd, which can be read independently of the discussion here.

Then in the sections Cohomology and Homotopy we describe, similarly leisurely, the intrinsic notions of cohomology and (geometric) homotopy in an (∞,1)-topos. Most aspects of what we say here involve fairly well-known facts, but the general abstract perspective of cohesive or at least ∞-connected (∞,1)-toposes seems to have not been fully appreciated before.

Finally in the section Differential cohomology we indicate how the combination of the intrinsic cohomology and geometric homotopy in a locally $\infty$-connected $(\infty,1)$-topos yields a good notion of differential cohomology in an (∞,1)-topos.

Toposes

There are several different perspectives on the notion of topos . One is that a topos is a category that looks like a category of spaces that sit by local homeomorphisms over a given base space: all spaces that are locally modeled on a given base space.

The archetypical class of examples are sheaf toposes $Sh(X) \simeq \mathrm{Et}(X)$ over a topological space $X$: these are the categories of etale spaces over $X$: topological spaces $Y$ that are equipped with a local homeomorphisms $Y \to X$. When $X = *$ is the point, this is just the category Set of all sets: spaces that are modeled on the point. This is the archetypical topos itself.

What makes the notion of toposes powerful is the following fact: even though the general topos contains objects that are considerably different from and possibly considerably richer than plain sets and even richer than etale spaces over a topological space, the general abstract category theoretic properties of every topos are essentially the same as those of $Set$. For instance in every topos all small limits and colimits exist and it is cartesian closed (even locally). This means that a large number of constructions in $Set$ have immediate analogs internal to every topos, and the analogs of the statements about these constructions that are true in $\mathrm{Set}$ are true in every topos.

This may be thought of as saying that toposes are very nice categories of spaces in that whatever construction on spaces one thinks of, the resulting space with the expected general abstract properties will exist in the topos. In this sense toposes are convenient categories for geometry ∞- as in: convenient category of topological spaces, but even more convenient than that.

On the other hand, we can de-emphasize the role of the objects of the topos and instead treat the topos itself as a “generalized space” (and in particular, a categorified space). We then consider the sheaf topos $\mathrm{Sh}(X)$ as a representative of $X$ itself, while toposes not of this form are “honestly generalized” spaces. This point of view is supported by the fact that the assignment $X \mapsto Sh(X)$ is a full embedding of (sufficiently nice) topological spaces into toposes, and that many topological properties of a space $X$ can be detected at the level of $\mathrm{Sh}(X)$.

Contrary to that, here we are mainly concerned with toposes that are far from being akin to sheaves over a topological space, and instead behave like abstract fat points with geometric structure . This implies that the objects of these toposes are in turn generalized spaces modeled locally on this geometric structure. Such toposes are called gros toposes or big toposes. There is a formalization of the properties of a topos that make it behave like a big topos of generalized spaces inside of which there is geometry: this is the notion of cohesive toposes.

More concretely, the idea of sheaf toposes formalizes the idea that any notion of space is typically modeled on a given collection of simple test spaces . For instance differential geometry is the geometry that is modeled on Cartesian spaces $\mathbb{R}^n$, or rather on the category $C =$ CartSp of Cartesian spaces and smooth functions between them.

A presheaf on such $C$ is a functor $X : C^{op} \to Set$ from the opposite category of $C$ to the category of sets.. We think of this as a rule that assigns to each test space $U \in C$ the set $X(U) :=: Maps(U,X)$ of structure-preserving maps from the test space $U$ into the would-be space $X$ - the probes of $X$ by the test space $U$. This assignment defines the generalized space $X$ modeled on $C$. Every category of presheaves over a small category is an example of a topos. But these presheaf toposes, while encoding the geometry of generalized spaces by means of probes by test spaces in $C$ fail to correctly encode the topology of these spaces. This is captured by restricting to sheaves among all presheaves.

Each test space $V \in C$ itself specifies a presheaf, by forming the hom-sets $Maps(U,V) := Hom_C(U,V)$ in $C$. This is called the Yoneda embedding of test spaces into the collection of all generalized spaces modeled on them. Presheaves of this form are the representable presheaves. A bit more general than these are the locally representable presheaves: for instance on $C =$ CartSp this are the smooth manifolds $X \in$ Diff whose presheaf-rule is $Maps(U,X) := Hom_{Diff}(U,X)$. By definition a manifold is locally isomorphic to a Cartesian space, hence is locally representable as a presheaf on CartSp.

These examples of presheaves on $C$ are special in that they are in fact sheaves: the value of $X$ on a test space $U$ is entirely determined by the restrictions to each $U_i$ in a cover $\{U_i \to U\}_{i \in I}$ of the test space $U$ by other test spaces $U_i$. We think of the subcategory of sheaves $Sh(C) \subset PSh(C)$ as consisting of those special presheaves that are those rules of probe-assignments which respect a certain notion of ways in which test spaces $U, V \in C$ may glue together to a bigger test space.

One may axiomatize this by declaring that the collections of all covers under consideration forms what is called a Grothendieck topology on $C$ that makes $C$ a site. But of more intrinsic relevance is the equivalent fact that categories of sheaves are precisely the subtoposes of presheaf toposes

$$Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C) = [C^op, Set] \,,$$

meaning that the embedding $Sh(X) \hookrightarrow PSh(X)$ has a left adjoint functor $L$ that preserves finite limits. This may be taken to be the definition of Grothendieck toposes. The left adjoint is called the sheafification functor. It is determined by and determines a Grothendieck topology on $C$.

For the choice $C =$ CartSp such is naturally given by the good open cover coverage, which says that a bunch of maps $\{U_i \to U\}_{i \in I}$ in $C$ exhibit the test object $U$ as being glued together from the test objects $\{U_i\}$ if these form a good open cover of $U$. With this notion of coverage every smooth manifold is a sheaf on $CartSp$.

But there are important genenralized spaces modeled on $CartSp$ that are not smooth manifolds: topological spaces for which one can consistently define which maps from Cartesian spaces into them count as smooth in a way that makes this assignment a sheaf on CartSp, but which are not necessarily locally isomorphic to a Cartsian space: these are called diffeological spaces. A central example of a space that is naturally a diffeological space but not a finite dimensional manifold is a mapping space $[\Sigma,X]$ of smooth functions between smooth manifolds $\Sigma$ and $X$: since the idea is that for $U$ any Cartesian space the smooth $U$-parameterized families of points in $[\Sigma,X]$ are smooth $U$-parameterized families of smooth maps $\Sigma \to X$, we can take the plot-assigning rule to be

$$[\Sigma, X] : U \mapsto Hom_{Diff}(\Sigma \times U , X) \,.$$

It is useful to relate all these phenomena in the topos $Sh(C)$ to their image in the archetypical topos Set. This is simply the category of sets, which however we should think of here as the category $Set \simeq Sh((*))$ of sheaves on the category $*$ which contains only a single object and no nontrivial morphism: objects in here are generalized spaces modeled on the point . All we know about them is how to map the point into them, and as such they are just the sets of all possible such maps from the point.

Every category of sheaves $Sh(C)$ comes canonically with an essentially unique geometric morphism to the topos of sets, given by a pair of adjoint functors

$$Sh(C) \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} Sh(*) = Set \,.$$

Here $\Gamma$ is called the global sections functor. If $C$ has a terminal object $*$, then it is given by evaluation on that object: the functor $\Gamma$ sends a plot-assigning rule $X : C^{op} \to Set$ to the set of plots by the point $*$: $\Gamma(X) = X(*)$. For instance in $C =$ CartSp the terminal object exists and is the ordinary point $* = \mathbb{R}^0$. If $X \in Sh(C)$ is a smooth manifold or diffeological space as above, then $\Gamma(X) \in Set$ is simply its underlying set of points. So the functor $\Gamma$ can be thought of as forgetting the cohesive structure that is given by the fact that our generalized spaces are modeled on $C$. It remembers only the underlying point-set.

Conversely, its left adjoint functor $Disc$ takes a set $S$ to the sheafification $Disc(S) = L(Const(S))$ of the constant presheaf $Const : U \mapsto S$, which asserts that the set of its plots by any test space is always the same set $S$. This is the plot-rule for discrete space modeled on $C$ given by the set $S$: a plot has to be a constant map of the test space $U$ to one of the elements $s \in S$. For the case $C = CartSp$ this interpretation is literally true in the familiar sense: the generalized smooth space $Disc(S)$ is the discrete smooth manifold or discrete diffeological space with point set $S$.

The examples for generalized spaces $X$ modeled on $C$ that we considered so far all had the property that the collection of plots $U \to X$ into them was a subset of the set of maps of sets from $U$ to their underlying set $\Gamma(X)$ of points. These are called concrete sheaves. Not every sheaf is concrete. The concrete sheaves form a subcategory inside the full topos which is itself almost, but not quite a topos: it is called the quasitopos of concrete objects

$$Conc(C) \stackrel{\leftarrow}{\hookrightarrow} Sh(C) \,.$$

Non-concrete sheaves over $C$ may be exotic as compared to smooth manifolds, but they are still usefully regarded as generalized spaces modeled on $C$. For instance for $n \in \mathbb{N}$ there is the sheaf $\kappa(n,\mathbb{R})$ given by saying that plots by $U \in CartSp$ are identified with closed differential n-forms on $U$:

$$\kappa(n,\mathbb{R}) : U \mapsto \Omega^n_{cl}(U) \,.$$

This sheaf describes a very non-classical space, which for $n \geq 1$ has only a single point, $\Gamma(\kappa(n,\mathbb{R})) = {*}$ , only a single curve, a single surface, etc., up to a single $(n-1)$-dimensional probe, but then it has a large number of $n$-dimensional probes. Despite the fact that this sheaf is very far in nature from the test spaces that it is modeled on, it plays a crucial and very natural role: it is in a sense a model for an Eilenberg-MacLane space $K(n,\mathbb{R})$. We shall see in Lie theory in an (∞,1)-topos that these sheaves are part of an incarnation of the L-∞-algebra $\mathbf{B}^n \mathbb{R}$ and the sense in which it models an Eilenberg-MacLane space is precisely that of Sullivan models in rational homotopy theory. In any case, we want to allow ourselves to regard non-concrete objects such as $\kappa(n, \mathbb{R})$ on the same footing as diffeological spaces and smooth manifolds.

$(\infty,1)$-Toposes

While therefore a general object in the sheaf topos $Sh(C)$ may exhibit a considerable generalization of the objects $U \in C$ that it is modeled on, for many natural applications this is still not quite general enough: if for instance $X$ is a smooth orbifold, then there is not just a set, but a groupoid of ways of probing it by a Cartesian test space $U$: if a probe $\gamma : U \to X$ is connected by an orbifold transformation to another probe $\gamma' : U \to X$, then this constitutes a morphism in the groupoid $X(U)$ of probes of $X$ by $U$.

Even more generally, there may be an entire ∞-groupoid of probes of the generalized space $X$ by the test space $U$: a set of probes with morphisms between different probes, 2-morphisms between these 1-morphisms, and so on.

Such structures are described in (∞,1)-category theory: where a category has a set of morphisms between any two objects, an $(\infty,1)$-category has an $\infty$-grouopoid of morphisms, whose compositions are defined up to higher coherent homotopy. The theory of $(\infty,1)$-categories is effectively the combination of category theory and homotopy theory. The main fact about it, emphasized originally by André Joyal and then further developed in HTT, is that it behaves formally entirely analously to category theory: there are notions of (∞,1)-functors,
(∞,1)-limits, adjoint (∞,1)-functors etc, that satisfy all the familiar relations from category theory. For instance right adjoint $(\infty,1)$-functors preserve all $(\infty,1)$-limits, there is an adjoint (∞,1)-functor theorem, an (∞,1)-Grothendieck construction-theorem, and so on.

In particular, there is a notion of (∞,1)-presheaves on a category (or $(\infty,1)$-category): $(\infty,1)$-functors

$$X : C^{\mathrm{op}} \to \infty \mathrm{Grpd}$$

to the $(\infty,1)$-category ∞Grpd of $\infty$-groupoids – there is an (∞,1)-Yoneda embedding, and so on. Accordingly, (∞,1)-topos theory proceeds in its basic notions along the same lines as we sketched above for topos theory:

an (∞,1)-topos of (∞,1)-sheaves is defined to be a reflective sub-(∞,1)-category

$$Sh_{(\infty,1)}(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh(C) := Funct_{(\infty,1)}(C^{op}, \infty Grpd)$$

of an (∞,1)-category of (∞,1)-presheaves.

As before, such is essentially determined by and determines a Grothendieck topology or coverage on $C$ (for this to be precise we need to demand that the inclusion is a topological localization).

Since a $(2,1)$-sheaf with values in groupoids is usually called a stack, a (∞,1)-sheaf is often also called an ∞-stack. The (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(C)$ is called an (∞,1)-topos. This is the kind of context of generalized spaces in which we shall develop our constructions.

Specifically, in the spirit of the above discussion, the objects of the $(\infty,1)$-topos of $(\infty,1)$-sheaves on $C =$ CartSp we shall think of as smooth ∞-groupoids. This is our main running example. We shall write $\mathrm{Smooth}\infty\mathrm{Grpd} := \mathrm{Sh}_\infty(\mathrm{CartSp})$ for the $\infty$-topos of smooth $\infty$-groupoids.

But a crucial point of developing our theory in the language of $(\infty,1)$-toposes is that all constructions work in great generality. By simply passing to another site $C$, all constructions apply to the theory of generalized spaces modeled on the test objects in $C$. Indeed, to really capture all aspects of $\infty$-Lie theory, we should and will adjoin to our running example $C =$ CartSp that of the slightly larger site $C =$ ThCartSp of infinitesimally thickened Cartesian spaces. Ordinary sheaves on this site are the generalized spaces considered in synthetic differential geometry: these are smooth spaces such as smooth loci that may have infinitesimal extension. For instance the first order jet $D \subset R$ of the origin in the real line exists as an infinitesimal space in $Sh(ThCartSp)$. Accordingly, ∞-groupoids modeled on ThCartSp are smooth ∞-groupoids that may have k-morphisms of infinitesimal extension. We will see that a smooth $\infty$-groupoid all whose morphisms has infinitesimal extension is a Lie algebra or Lie algebroid or generally a ∞-Lie algebroid.

While (∞,1)-category theory provides a good abstract definition and theory of $\infty$-groupoids modeled on test objects in a category $C$ in terms of the (∞,1)-category of (∞,1)-sheaves on $C$, for concrete manipulations it is often useful to have a presentation of the (∞,1)-categories in question in terms of generators and relations in ordinary category theory. Such a generators-and-relations presentation is provided by the notion of a model category. Specifically, the $(\infty,1)$-toposes of $(\infty,1)$-presheaves that we are concerned with are presented in this way by a model structure on simplicial presheaves, i.e. on the functor category $[C^{op}, sSet]$ from $C$ to the category sSet of simplicial sets.

In terms of this model, the corresponding (∞,1)-category of (∞,1)-sheaves is given by another model structure on $[C^{op}, sSet]$, called the left Bousfield localization at the set of covers in $C$.

These models for ∞-stack (∞,1)-toposes have been proposed, known and studied since the 1970s and are therefore quite well understood. The full description and proof of their abstract role in higher category theory was established in Lurie, HTT.

Generators-and-relations presentation for $(\infty,1)$-toposes

$$\array{ \mathbf{H} := & Sh_{(\infty,1)}(C) & \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} & PSh_{(\infty,1)}(C) && \text{intrinsic def of (∞,1)-topos} \\ & \uparrow\mathrlap{\simeq} && \uparrow\mathrlap{\simeq} && \text{Lurie's theorem} \\ & ([C^{op}, sSet]_{proj,cov})^\circ &\stackrel{\stackrel{\text{left Bousfield loc.}}{\leftarrow}}{\to}& ([C^{op}, sSet]_{proj})^\circ && \text{model by simplicial presheaves} }$$

As before for toposes, there is a canonical (∞,1)-topos, which is ∞Grpd = $Sh_{(\infty,1)}(*)$ itself: the collection of generalized ∞-groupoids that are modeled on the point. All we know about these generalized spaces is how to map a point into them and what the homotopies and higher homotopies of such maps are, but no further extra structure. So these are bare ∞-groupoids without extra structure. Also as before, every (∞,1)-topos comes with an essentially unique geometric morphism to this archetypical $(\infty,1)$-topos given by a pair of adjoint (∞,1)-functors.

$$Sh_{(\infty,1)}(C) \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.$$

Again, if $C$ happens to have a terminal object $*$, then $\Gamma$ is the operation that evaluates an $(\infty,1)$-sheaf on the point: it produces the bare $\infty$-groupoid underlying an $\infty$-groupoid $X \in Sh_{(\infty,1)}(C)$ modeled on $C$. For instance for $C =$ CartSp an smooth ∞-groupoid $X \in Sh_{(\infty,1)}(C)$ is sent by $\Gamma$ to to the underlying $\infty$-groupoid that forgets the smooth structure on $X$.

Moreover, still in direct analogy to the 1-categorical case above, the left adjoint $Disc$ is the the (∞,1)-functor that sends a bare ∞-groupoid $S$ to the ∞-stackification $L(Const(S))$ of the constant $(\infty,1)$-presheaf $Const S : U \mapsto S$. This models the discretely structured $\infty$-groupoid on $S$. For instance for $C = CartSp$ we have that $LConst S$ is a smooth $\infty$-groupoid with discrete smooth structure: all smooth families of points in it are actually constant.

Cohomology

We had mentioned that every topos behaves in most general abstract ways as the archetypical topos $Set$. Analogously, every $(\infty,1)$-topos behaves in most general abstract ways as the archetypical $(\infty,1)$-topos ∞Grpd. This, in turn, by the homotopy hypothesis-theorem, is equivalent to Top, the category of topological spaces, regarded as an (∞,1)-category by taking the 2-morphisms to be homotopies between continuous maps, 3-morphisms to be homotopies of homotopy, and so forth. The equivalence

$$(|-| \dashv \Pi = Sing) : Top \stackel{\leftarrow}{\overset{\simeq}{\to}} \infty Grpd$$

is constituted by forming fundamental ∞-groupoids of topological spaces modeled by forming singular simplicial complexes and by forming geometric realization of simplicial sets.

In Top it is familiar – from the notion of classifying space and the Brown representability theorem in particular – that the cohomology of a topological space $X$ is defined as the set of homotopy classes of maps from $X$ to some coefficient space $A$

$$H(X,A) := \pi_0 Top(X,A) \,.$$

For instance for $A = K(n,\mathbb{Z})$ an Eilenberg-MacLane space, we have that

$$H(X,A) = \pi_0 Top(X, K(\mathbb{Z},n)) \simeq H^n(X,\mathbb{Z})$$

is the ordinary integral singular cohomology of $X$. Also nonabelian cohomology is modeled this way: for $G$ a (possibly nonabelian ) topological group and $A = \mathcal{B}G$ its classifying space we have that

$$H(X,A) = \pi_0 Top(X,B G) \simeq H^1(X,G)$$

is the degree-1 nonabelian cohomology of $X$ with coeffients in $G$, which classifies $G$-principal bundles on $X$.

Since this only involves forming (∞,1)-categorical hom-spaces and since this is an entirely categorical operation, it makes sense to define for $X, A$ two objects in an arbitrary $(\infty,1)$-topos $\mathbf{H}$ the intrinsic cohomology of $X$ with coefficients in $A$ to be

$$H(X,A) := \pi_0 \mathbf{H}(X,A) \,,$$

where $\mathbf{H}(X,A)$ denotes the $\infty$-groupoid of morphisms from $X$ to $A$ in $\mathbf{H}$.

It turns out that essentially every notion of cohomology considered in the literature is an example of this simple definition, for a suitable choice of $\mathbf{H}$. Notably abelian sheaf cohomology over a given site $C$ is the special case where $\mathbf{H} = Sh_{(\infty,1)}(C)$ and $A$ takes values in abelian simplicial groups. This example alone subsumes a wealth of further special cases, such as for instance Deligne cohomology.

There are some definitions in the literature of cohomology theories that are not special cases of this general concept, but in these cases it seems that the failure is with the traditional definition, not with the above notion. We shall be interested in particular in the group cohomology of Lie groups.

Originally this was defined using a naive direct generalization of the formula for bare group cohomology as

$$H^n_{naive}(G,A) = \{smooth maps G^{\times n} \to A\}/\sim \,.$$

But this definition was eventually found to too coarse: there are structures that ought to be cocycles on Lie groups but do not show up in this definition. Graeme Segal therefore proposed a refined definition that was later rediscovered by Jean-Luc Brylinski, called differentiable Lie group cohomology $H^n_{diffbl}(G,A)$. This refines the naive Lie group cohomology in that there is a natural morphism $H^n_{naive}(G,A) \to H^n_{diffbl}(G,A)$.

But in the (∞,1)-topos of smooth ∞-groupoids $\mathbf{H} = Sh_{(\infty,1)}(CartSp)$ we have also the natural intrinsic definition of Lie group cohomology as

$$H^n_{\mathrm{Smooth}}(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A) \,,$$

where the boldface $\mathbf{B}$ denotes the intrinsic delooping in $\mathbf{H}$. We find that generally this naturally includes the Segal/Brylinski definition

$$H^n_{naive}(G,A) \to H^n_{diffrbl}(G,A) \to H^n_{Smooth}(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)$$

and at least for $A$ a discrete group, or the group of real numbers or a quotient of these such as $U(1) = \mathbb{R}/\mathbb{Z}$, the notions coincide

$$H^n_{diffrbl}(G,A) \simeq H^n_{Smooth}(G,A) \,.$$

This general abstract reformulation of Lie group cohomology in $(\infty,1)$-topos theory allows to deduce some properties of it in great generality. For instance one of the crucial aspects of the notion of cohomology is that a cohomology class on $X$ classifies certain structures over $X$.

It is a classical fact that if $G$ is a (discrete) group and $\mathcal{B}G$ its delooping in Top, then the structure classified by a cocycle $g : X \to \mathcal{B}G$ is the $G$-principal bundle over $X$ obtained as the 1-categorical pullback $P \to X$

$$\array{ P &\to& \mathcal{E} G \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}G }$$

of the universal principal bundle $\mathcal{E}G \to \mathcal{B}G$ over $G$. But one finds that this pullback construction is just a 1-categorical model for what intrinsically is something simpler: this is just the homotopy pullback in Top of the point

$$\array{ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}G } \,.$$

This form of the construction of the $G$-principal bundle classified by a cocycle makes sense in any (∞,1)-topos $\mathbf{H}$:

we shall say that for $G \in \mathbf{H}$ a group object in $\mathbf{H}$ and $\mathbf{B}G$ its delooping and for $g : X \to \mathbf{B}G$ a cocycle, i.e. just a morphism in $\mathbf{H}$, that the $G$-principal ∞-bundle classified by $g$ is the (∞,1)-categorical pullback

$$\array{ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }$$

in $\mathbf{H}$.

Let $G$ be a Lie group and $X$ a smooth manifold, both regarded naturally as objects in the $(\infty,1)$-topos of smooth ∞-groupoids. Let $g : X \to \mathbf{B}G$ be a morphism in $\mathbf{H}$. One finds that in terms of the presentation of $Smooth\infty Grpd$ by the model structure on simplicial presheaves this is a Cech 1-cocycle on $X$ with values in $G$. The corresponding $(\infty,1)$-pullback $P = X \times_{\mathbf{B}G} *$ is (up to equivalence or course) the smooth $G$-principal bundle classified in the usual sense by this cocycle.

The analogous proposition holds for $G$ a Lie 2-group and $P$ a $G$-principal 2-bundle.

Generally, we can give a natural definition of $G$-principal ∞-bundle in any $(\infty,1)$-topos $\mathbf{H}$ over any group object $G \in \mathbf{H}$. One finds that it is the Giraud axioms that characterize (∞,1)-toposes that ensure that these are equivalently classified as the $(\infty,1)$-bullbacks of morphisms $g : X \to \mathbf{B}G$. Therefore the intrinsic cohomology

$$H(X,G) := \pi_0\mathbf{H}(X,\mathbf{B}G)$$

in $\mathbf{H}$ classifies $G$-principal ∞-bundles over $X$. Notice that $X$ itself here may be any object in $\mathbf{H}$.

Homotopy

Every (∞,1)-sheaf (∞,1)-topos $\mathbf{H}$ canonically comes equipped with a geometric morphism given by pair of adjoint (∞,1)-functors

$$(L Const \dashv \Gamma) : \mathbf{H} \stackrel{\overset{L Const}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$$

relating it to the archeytpical $(\infty,1)$-topos of ∞-groupoids. Here $\Gamma$ produces the global sections of an (∞,1)-sheaf and $L Const$ produces the constant ∞-stack on a given ∞-groupoid.

In the cases that we are interested in here $\mathbf{H}$ is a big topos of ∞-groupoids equipped with cohesive structure , notably equipped with smooth structure in our motivating example. In this case $\Gamma$ has the interpretation of sending a cohesive $\infty$-groupoid $X \in \mathbf{H}$ to its underlying $\infty$-groupoid, after forgetting the cohesive structure, and $L Const$ has the interpretation of forming $\infty$-groupoids equipped with discrete cohesive structure. We shall write $Disc := L Const$ to indicate this.

But in these cases of cohesive (∞,1)-toposes there are actually more adjoints to these two functors, and this will be essentially the general abstract definition of cohesiveness. In particular there is a further left adjoint

$$\Pi : \mathbf{H} \to \infty Grpd$$

to $Disc$: the fundamental ∞-groupoid functor on a locally ∞-connected (∞,1)-topos. Following the standard terminology of locally connected toposes in ordinary topos theory we shall say that $\mathbf{H}$ with such a property is a locally ∞-connected (∞,1)-topos. This terminology reflects the fact that if $X$ is a locally contractible topological space then $\mathbf{H} = Sh_{(\infty,1)}(X)$ is a locally contractible $(\infty,1)$-topos. A classical result of Artin-Mazur implies, that in this case the value of $\Pi$ on $X \in Sh_{(\infty,1)}(X)$ is, up to equivalence, the fundamental ∞-groupoid of $X$:

$$\Pi : ((X \in Sh_{(\infty,1)}(X)) \mapsto (Sing X \in \infty Grpd)) \,,$$

which is the $\infty$-groupoid $\Pi X$ whose

• objects are the points of $X$;

• morphisms are the (continuous) paths in $X$;

• 2-morphisms are the continuous homotopies between such paths;

• k-morphisms are the higher order homotopies between $(k-1)$-dimensional paths.

This is the object that encodes all the homotopy groups of $X$ in a canonical fashion, without choice of fixed basepoint.

Also the big (∞,1)-topos Smooth∞Grpd $= Sh_{(\infty,1)}(CartSp)$ turns out to be locally $\infty$-connected

$$(\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} = Smooth\infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd$$

as a reflection of the fact that every Cartesian space $\mathbb{R}^n \in$ CartSp is contractible as a topological space. We find that for $X$ any paracompact smooth manifold, regarded as an object of Smooth∞Grpd, again $\Pi(X) \in \infty Grpd$ is the corresponding fundamental ∞-groupoid. More in detail, under the homotopy hypothesis-equivalence $Top \stackrel{\overset{|-|}{\leftarrow}}{\underoverset{Sing}{\simeq}{\to}} \infty Grpd$ we have that the composite

$$|\Pi(-)| : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{|-|}{\to} Top$$

sends a smooth manifold $X$ to its homotopy type: the underlying topological space of $X$, up to weak homotopy equivalence.

Analogously, for a general object $X \in \mathbf{H}$ we may think of $|\Pi(X)|$ as the generalized geometric realization in Top. For instance we find that if $X \in \infty Lie Grpd$ is presented by a simplicial paracompact manifold, then $|\Pi(X)|$ is the ordinary geometric realization of the underlying simplicial topological space of $X$. This means in particular that for $X \in Smooth\infty Grpd$ a Lie groupoid, $\Pi(X)$ computes its homotopy groups of a Lie groupoid as traditionally defined.

The ordinary homotopy groups of $\Pi(X)$ or equivalently of $|\Pi(X)|$ we call the geometric homotopy groups of $X \in \mathbf{H}$, because these are based on a notion of homotopy induced by an intrisic notion of geometric paths in objects in $X$. This is to be contrasted with the categorical homotopy groups of $X$. These are the homotopy groups of the underlying $\infty$-groupoid $\Gamma(X)$ of $X$. For instance for $X$ a smooth manifold we have that

$$\pi_n(\Gamma(X)) \simeq \left\{ \array{ X \in Set & | n = 0 \\ 0 & | n \gt 0 } \right.$$

but

$$\pi_n(\Pi(X)) \simeq \pi_n(X \in Top) \,.$$

This allows us to give a precise sense to what it means to have a cohesive refinement (continuous refinement, smooth refinement, etc.) of an object in $Top$. Notably we are interested in smooth refinements of classifying spaces $B G \in Top$ for topological groups $G$ by deloopings $\mathbf{B}G \in \infty Lie Grpd$ of ∞-Lie groups $G$ and we may interpret this as saying that

$$\Pi(\mathbf{B}G) \simeq B G$$

in $Top \simeq \infty Grpd$.

Differental cohomology

We now indicate how the combination of the intrinsic cohomology and the geometric homotopy in a locally ∞-connected (∞,1)-topos yields a good notion of differential cohomology in an (∞,1)-topos.

Using the defining adjoint (∞,1)-functors $(\Pi \dashv \Disc \dashv \Gamma)$ we may reflect the fundamental ∞-groupoid $\Pi : \mathbf{H} \to \infty Grpd \simeq Top$ from Top back into $\mathbf{H}$ by considering the composite endo-edjunction

$$(\mathbf{\Pi} \dashv \mathbf{\flat}) := (Disc \circ \Pi \dashv Disc \circ \Gamma) : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} \,.$$

The $(\Pi \dashv Disc)$-unit $X \to \mathbf{\Pi}(X)$ may be thought of as the inclusion of $X$ into its fundamental $\infty$-groupoid as the collection of constant paths in $X$.

As always, the boldface $\mathbf{\Pi}$ is to indicate that we are dealing with a cohesive refinement of the topological structure $\Pi$. The symbol “$\mathbf{\flat}$” (“flat”) is to be suggestive of the meaning of this construction:

since for $X \in \mathbf{H}$ any cohesive object, we may think of $\mathbf{\Pi}(X)$ as its cohesive fundamental $\infty$-groupoid, a morphism

$$\nabla : \mathbf{\Pi}(X) \to \mathbf{B}G \,,$$

(hence a $G$-valued cocycle on $\mathbf{\Pi}(X)$) may be interpreted as assigning:

• to each point in $X$ the fiber of the corresponding $G$-principal ∞-bundle classified by the composite $g : X \to \mathbf{\Pi}(X) \stackrel{\nabla}{\to} \mathbf{B}G$;

• to each path in $X$ an equivalence between the fibers over its endpoints;

• to each homotopy of paths in $X$ an equivalence between these equivalences;

• and so on.

This in turn we may think as being the flat higher parallel transport of an ∞-connection on the bundle classified by $X \to \mathbf{\Pi}(X) \to \mathbf{B}G$.

The adjunction equivalence allows us to identify $\mathbf{\flat} \mathbf{B}G$ as the coefficient object for this flat differential $G$-valued cohomology on $X$:

$$H_{flat}(X,G) := \pi_0 \mathbf{H}(X, \mathbf{\flat} \mathbf{B}G) \simeq \pi_0 \mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}G) \,.$$

In $\mathbf{H} =$ Smooth∞Grpd and with $G \in \mathbf{H}$ an ordinary Lie group and $X \in \mathbf{H}$ an ordinary smooth manifold, we have that $H_{flat}(X, G)$ is the set of equivalence classes of ordinary $G$-principal bundles on $X$ with flat connections.

The $(\Disc \dashv \Gamma)$-counit $\mathbf{\flat} \mathbf{B}G \to \mathbf{B}G$ provides the forgetful map

$$H_{flat}(X,G) \to H(X, G)$$

form $G$-principal $\infty$-bundles with flat connection to their underlying principal $\infty$-bundles. Not every $G$-principal $\infty$-bundle admits a flat connection. The failure of this to be true – hence the obstruction theory to flat lifts – is measured by the homotopy fiber of the counit, which we shall denote $\mathbf{\flat}_{dR} \mathbf{B}G$, defined by the fact that we have a fiber sequence

$$\mathbf{\flat}_{dR} \mathbf{B}G \to \mathbf{\flat} \mathbf{B}G \to \mathbf{B}G \,.$$

As the notation suggests, it turns out that $\mathbf{\flat}_{dR} \mathbf{B}G$ may be thought of as the coefficient object for nonabelian generalized de Rham cohomology. For instance for $G$ an odinary Lie group regarded as an object in $\mathbf{H} = \infty Lie Grpd$, we have that $\mathbf{\flat}_{dR} \mathbf{B}G$ is presented by the sheaf $\Omega_{flat}^1(-, \mathfrak{g})$ of Lie algebra valued differential forms with vanishing curvature 2-form. And for the circle Lie n-group $\mathbf{B}^{n-1} U(1)$ we find that $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is presented by the complex of sheaves whose abelian sheaf cohomology is de Rham cohomology in degree $n$

$$H(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1)) \simeq H_{dR}^n(X) \,.$$

(More precisely, this is true for $n \geq 2$. For $n = 1$ we get just the sheaf of closed 1-forms. This is due to the obstruction-theoretic nature of $\mathbf{\flat}_{\mathrm{dR}}$: as we shall see, in degree 1 it computes 1-form curvatures of groupoid principal bundles, and these are not quotiented by exact 1-forms.)

Moreover, in this case our fiber sequence extends not just to the left but also to the right

$$\mathbf{\flat}_{dR} \mathbf{B}^n U(1) \to \mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) \,.$$

The induced morphism

$$\mathbf{H}(X, \mathbf{B}^n U(1)) \to \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1))$$

we may think of as equipping an $\mathbf{B}^{n-1} U(1)$-principal n-bundle (equivalently an $(n-1)$-[nLab:bundle gerbe]) with a connection, and then sending it to the higher curvature class of this connection.

The homotopy fibers

$$\mathbf{H}_{diff}(X,\mathbf{B}^n U(1)) \to \mathbf{H}(X, \mathbf{B}^n U(1)) \to \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1))$$

of this map therefore have the interpretation of being the cocycle $\infty$-groupoids of circle n-bundles with connection. This is the realization in Smooth∞Grpd of the general definition of ordinary differential cohomology in a cohesive $(\infty,1)$-topos.

All these definitions make sense in full generality for any locally ∞-connected (∞,1)-topos. We used nothing but the existence of the triple of adjoint (∞,1)-functors $(\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \to \infty Grpd$. We shall show for the special case that $\mathbf{H} =$ Smooth∞Grpd and $X$ an ordinary smooth manifold, that this general abstract definition reproduces ordinary differential cohomology over smooth manifolds as traditionally considered.

The advantage of the general abstract reformulation is that it generalizes the ordinary notion naturally to base objects that may be arbitrary smooth ∞-groupoids. This gives in particular the ∞-Chern-Weil homomorphism in an almost tautological form:

for $G \in \mathbf{H}$ any ∞-group object and $\mathbf{B}G \in \mathbf{H}$ its delooping, we may think of a morphism

$$\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$$

as a representative of a characteristic class on $G$, in that this induces a map

$$[\mathbf{c}(-)] : H(X,G) \to H^n(X, U(1))$$

from $G$-principal ∞-bundles to degree-$n$ cohomology-classes. Since the classification of $G$-principal $\infty$-bundles by cocycles is entirely general, we may equivalently think of this as the $\mathbf{B}^{n-1}U(1)$-principal $\infty$-bundle $P \to \mathbf{B}G$ given as the homotopy fiber of $\mathbf{c}$. A famous example is the Chern-Simons circle 3-bundle for $G$ a simply connected Lie group.

By postcomposing further with the canonical morphism $\mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)$ this gives in total a differential characteristic class

$$\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$$

that sends a $G$-principal ∞-bundle to a class in de Rham cohomology

$$[\mathbf{c}_{dR}] : H(X, G) \to H_{dR}^{n+1}(X) \,.$$

This is the generalization of the plain Chern-Weil homomorphism.associated with the characteristic class $c$. In cases accessible by traditional theory, it is well known that this may be refined to what are called the assignment of secondary characteristic classes to $G$-principal bundles with connection, taking values in ordinary differential cohomology

$$[\hat \mathbf{c}] : H_{conn}(X, G) \to H_{diff}^{n+1}(X) \,.$$

We show that in the general abstract formulation this corresponds to finding the universal object $\mathbf{B}G_{conn}$ that lifts all curvature characteristic classes to their corresponding circle n-bundles with connection, in that it fits into the diagram

$$\array{ \mathbf{H}(-,\mathbf{B}G_{conn}) &\to& \prod_i \mathbf{H}_{diff}(-, \mathbf{B}^{n_i} U(1)) &\to& \prod_i H_{dR}^{n_i+1}(-) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(-, \mathbf{B}G) &\to& \prod_i \mathbf{H}(-, \mathbf{B}^{n_i} U(1)) &\stackrel{curv}{\to}& \mathbf{H}(-, \flat_{dR}\mathbf{B}^{n_i+1}U(1)) } \,.$$

The cocycles in $\mathbf{H}_{conn}(X,\mathbf{B}G) := \mathbf{H}(X,\mathbf{B}G_{conn})$ we may identify with ∞-connections on the underlying principal ∞-bundles. Specifically for $G$ an ordinary Lie group this captues the ordinary notion of connection on a bundle, for $G$ Lie 2-group it captures the notion of connection on a 2-bundle/gerbe.

Implementation on smooth $\infty$-groupoids

• ∞-Chern-Weil theory introduction.

Applications

The generalized (nonabelian/smooth/differential) cohomology that we consider here has applications to – and is to a good degree motivated from – the study of backgrounds for 2-dimensional conformal field theory sigma-models. Before stating some of our results in this context, we briefly reviêw the context of these considerations.

In perturbative string theory one studies a generalization of Feynman perturbation series, which describes the scattering processes of quantum objects as observed in a particle accelerator, from a sum over graphs of correlators of a 1-dimensional quantum field theory on these graphs, to a sum over Riemann surfaces of correlators of a 2-dimensional conformal quantum field theory on these surfaces.

The choice of 2-dimensional CFT used in this series determines and is determined by the spacetime in which the quantum objects whose scattering is being described by the series propagate, including the configuration of the field of gravity and of gauge fields present in this spacetime that exert forces on these quantum objects while they scatter wiith each other.

The mathematical formalization of what all this means is not completed yet, but considerable progress has been achieved in recent years. For a survey see Mathematical Foundations of Quantum Field and Perturbative String Theory. One central insight has been that the background spacetime with its gauge fields is mathematically to be modeled by a cocycle in generalized differential cohomology. Not every cocycle however corresponds under the above correspondence to a suitable 2-dimensional CFT. While the process that connects background field data to the corresponding sigma-model quantum field theory – which is the quantization of the sigma-model in this background – is not completely formalized yet (not the least because 2-dimensional CFTs themselves are formally under complete control only in special cases) a few nontrivial consistency conditions on the background field data are known and understood: the quantum anomaly-cancellation conditions that encode the vanishing of certain obstructions for the sigma-model action functional to be a function on configuration space, instead of a section of a nontrivial line bundle. The quantum anomaly is the nontriviality of this bundle on configuration space, and the non-flatness of its connection.

There are two main types of such anomalies. One is induced from the presence of fermions in the theory. The partial path integral over these field yields a Pfaffian line bundle on the remaining configuration space of the bosonic fields, which need not be trivial. The other is the presence of higher background magnetic charge for the given higher gauge fields.

A consistent sigma-model is obtained if both these anomalies vanish separately. But they may also be each nontrivial, but inverse to each other, so that they cancel out. This cancelling of fermionic with higher gauge theoretic quantum anomalies is known as a Green-Schwarz mechanism.

It turns out that this involves an interplay between generalized (Eilenberg-Steenrod) differential cohomology and differential nonabelian cohomology. This situation can be described with the machinery of differential cohomology in an $(\infty,1)$-topos described here.

(see SSSI, SSSII and SSSIII)

For the path integral quantization of the sigma-model for the super-$n$-brane propating on $X$ to make sense (at all) a certain obstruction called a quantum anomaly has to vanish, which imposes the following constraints on the cohomology of $X$, i.e. on the kinematics of the physical system:

• for $n= 0$, we need to require an orientation on $X$: a lift

  $$\array{ && B SO(k) &&&& cokill \pi_{0+1} \\ & \nearrow & \downarrow \\ X &\to& B O(k) }$$

• for $n= 1$, we need to require a Spin structure on $X$: a lift

  $$\array{ && B Spin(k) &&&& cokill \pi_{1+1} \\ & \nearrow & \downarrow \\ X &\to& B SO(k) }$$

• for $n= 2$, we need to require a String structure on $X$: a lift

  $$\array{ && B String(k)&&&& cokill \pi_{3+1} \\ & \nearrow & \downarrow \\ X &\to& B Spin(k) }$$

• for $n= 6$, we think we need to require a Fivebrane structure on $X$: a lift

  $$\array{ && B Fivebrane(k) &&&& cokill \pi_{7+1} \\ & \nearrow & \downarrow \\ X &\to& B String(k) }$$

Here

$$\cdots B Fivebrane(k) \to B String(k) \to B Spin(k) \to B SO(k) \to B O(k)$$

is the Whitehead tower of $B O(n)$: each step towards the left gives the universal next higher connected cover of the previous space.

This allows us to identify the Green-Schwarz mechanism for the heterotic string with the existence of a connection on a twisted string 2-group-principal 2-bundle. And the GS-mechanism for the dual heterotic string, the Fiverbrane, with the existence of a connection on a twisted fivebrane 6-group-principal 6-bundle.

A key ingredient for making sense of this is the following result about smooth refinements of the Whitehead tower of the orthogonal group.

Proposition

(smooth refinement of Whitehead tower of $O(n)$)

In the (∞,1)-topos of smooth ∞-groupoids $\mathbf{H} = Sh_{(\infty,1)}(CartSp)$ the normalized canonical Lie algebra cocycles $\mu_3, \mu_7 \in CE(\mathfrak{so}(n))$ integrate, respectively, to a cocycle

$$\int \mu_3 : \mathbf{B}Spin(n) \to \mathbf{B}^3 U(1)$$

in $\mathbf{H}$ classifying an extension

$$\mathbf{B}U(1) \to String(n) \to Spin(n)$$

and a cocycle

$$\int \mu_7 : \mathbf{B}String(n) \to \mathbf{B}^7 U(1)$$

classifying an extension

$$\mathbf{B}^6 U(1) \to Fivebrane(n) \to String(n)$$

in $\mathbf{H}$, such that their image in $\infty Grpd \simeq Top_{cg,wH}$ are the 3- and 7-connected universal cover of the topological group $Spin(n)$, respectively.

$$\array{ |\Pi(Fivebrane)| &\simeq& O \langle 7\rangle \\ \downarrow && \downarrow \\ |\Pi(String)| &\simeq& O \langle 3\rangle \\ \downarrow && \downarrow \\ |\Pi(Spin)| &\simeq& O \langle 1\rangle \\ \downarrow && \downarrow \\ |\Pi(S O)| &\simeq& O \langle 0\rangle \\ \downarrow && \downarrow \\ |\Pi(O)| &\simeq& O }$$

Specifically, $String(d) \in \mathbf{H}$ is equivalent to the strict ∞-Lie groupoid

$$String(d) \simeq \left( P_* Spin(d) \times \hat \Omega_* Spin(n) \stackrel{\overset{\rho}{\to}}{\underset{p_2}{\to}}P_* Spin(d) \right)$$

where

• $P_* Spin(d)$ is the based path space of Spin(d);

• $\hat \Omega_* Spin(d)$ is the level-1 Kac-Moody central extension of the based loop group of $Spin(d)$.

• $\rho$ is the evident action of loops on paths

  (actually there are three evident choices, they all yield equivalent strict models).

Proof

See String 2-group and Fivebrane 6-group.

2.) General abstract theory

We discuss a general abstract theory of (∞,1)-toposes that serve as contexts for higher geometry of cohesive ∞-groupoids.

In (∞,1)-Toposes we consider some basic notions of (∞,1)-topos theory to set up our context and notation. In Cohesive (∞,1)-toposes we consider axiomatics of big $(\infty,1)$-toposes. These induce a wealth of general abstract internal structures, which we list and discuss in Structures in a cohesive (∞,1)-topos. In Infinitesimal cohesion-topos#InfinitesimalCohesion) we add one more axiom that characterizes infinitesimal cohesive structure and again discuss the induced structures.

$(\infty,1)$-Toposes

We assume now that the reader is familiar with a few basic notions in (HTT). A route through the material that we do refer to is given at Seminar on (∞,1)-Categories and ∞-Stacks. A useful meta-theorem to keep in mind, originally emphasized by André Joyal and Charles Rezk is

Fact

• (∞,1)-Category theory parallels category theory.

• (∞,1)-Topos theory parallels topos theory.

This means that essentially all the standard constructions and theorems have their $\infty$-analogs if only we replace isomorphism consistently by equivalence in an (∞,1)-category. Notably there is the evident notion of (∞,1)-category of (∞,1)-presheaves and an (∞,1)-Yoneda lemma applies to it. There is a notion of (∞,1)-limits and adjoint (∞,1)-functors and they satisfy the expected properties. This allows to define (∞,1)-geometric morphisms as certain pairs of adjoint $(\infty,1)$-functors and finally to define (∞,1)-sheaf (∞,1)-toposes as (∞,1)-geometric embeddings inside (∞,1)-categories of (∞,1)-presheaves.

Following (HTT), for us “$(\infty,1)$-topos” means this:

Definition

An (∞,1)-sheaf (∞,1)-topos is an acessible (∞,1)-geometric embedding

$$\mathbf{H} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$$

into an (∞,1)-category of (∞,1)-presheaves over some small (∞,1)-category $C$.

We say this is an (∞,1)-category of (∞,1)-sheaves (as opposed to a hypercompletion of such) if $\mathbf{H}$ is the reflective localization at the covering sieves of a Grothendieck topology on the homotopy category of $C$ (a topological localization), and then write $\mathbf{H}= Sh_{(\infty,1)}(C)$ (with the site structure on $C$ understood).

For $\mathbf{H}$ an $(\infty,1)$-topos we write $\mathbf{H}(X,Y)$ for its ∞-hom ∞-groupoid between objects $X$ and $Y$, and write $H(X,Y) = \pi_0 \mathbf{H}(X,Y)$ for the hom-set in the homotopy category of an (∞,1)-category.

(So we do not consider a more general notion of “elementary $(\infty,1)$-topos”.)

Our discussion revolves around situations where the the following fact has a refinement:

Proposition

For every $(\infty,1)$-topos $\mathbf{H}$ there is an essentially unique (∞,1)-geometric morphism to the $(\infty,1)$-topos ∞Grpd.

$$(\Delta \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$$

This is (HTT, prop. 6.3.41). Here $\Gamma$ takes global sections and $\Delta$ forms constant (∞,1)-sheaves.

For computations it is useful to employ a generators and relations presentation for presentable (∞,1)-categories in general and (∞,1)-toposes in particular, given by ordinary sSet-enriched categories equipped with the structure of combinatorial simplicial model categories. These may be obtained by left Bousfield localization of a model structure on simplicial presheaves. (See appendix 2 and 3 of HTT.) Specifically we have the following.

Proposition

Let $C$ be a site (regarded as a small category with a coverage (Elephant)). Write

• $\mathrm{Sh}_{(\infty,1)}(C)$ for the (∞,1)-category of (∞,1)-sheaves over $C$.

• $[C^{op}, \mathrm{sSet}]_{\mathrm{proh},\mathrm{loc}}^{\circ}$ for the full subcategory on the fibrant-cofibrant objects of the left Bousfield localization of the projective simplicial model category model structure on simplicial presheaves over $C$ at the set of Cech nerve projections $C(\{U_i\}) \to U$ for each covering family $\{U_i \to U\}$ in $C$.

There is an equivalence of (∞,1)-categories

$$\mathrm{Sh}_{(\infty,1)}(C) \simeq [C^{op}, \mathrm{sSet}]_{\mathrm{proj}, \mathrm{loc}}^{\circ} \,.$$

Proof

By the discussion in the appendix of (http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+–+references#FSS) we have that the $(\infty,1)$-category presented by the left Bousfield localization at the coverage is equivalent to that obtained by left Bousfield localization at the full Grothendieck topology that it generates.

The statement then follows as in the proof of HTT prop. 6.5.2.14.

See model structure on simplicial presheaves – Localization and descent for details.

In terms of this presentation of $(\infty,1)$-categories, adjoint (∞,1)-functors are presented by simplicial Quillen adjunctions between simplicial model categories: the restriction of a simplicial Quillen adjunction to fibrant-cofibrant objects is the sSet-enriched functor that presents the $(\infty,1)$-derived functor under the model of (∞,1)-categories by simplicially enriched categories.

Proposition

Let $C$ and $D$ be simplicial model categories and let

$$(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$$

be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let $C^\circ$ and $D^\circ$ be the (∞,1)-categories presented by $C$ and $D$ (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

$$(\mathbb{L}L \dashv \mathbb{R}R) : C^\circ \stackrel{\leftarrow}{\to} D^{\circ} \,.$$

On the decategorified level of the homotopy categories these are the total left and right derived functors, respectively, of $L$ and $R$.

This is (HTT, prop 5.2.4.6).

The following proposition states conditions under which a simplicial Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

Proposition

If $C$ and $D$ are simplicial model categories and $D$ is a left proper model category, then for an sSet-enriched adjunction

$$(L \dashv R) : C \stackrel{\leftarrow}{\to} D$$

to be a Quillen adjunction it is already sufficient that $L$ preserves cofibrations and $R$ preserves fibrant objects.

This appears as (HTT, cor. A.3.7.2).

Remark

We will use this for finding simplicial Quillen adjunctions into left Bousfield localizations of left proper model categories: the left Bousfield localization preserves the left properness, and the fibrant objects in the Bousfield localized structure have a good characterization: they are the fibrant objects in the original model structure that are also local objects with respect to the set of morphisms at which one localizes.

Therefore for $D$ the left Bousfield localization of a simplicial left proper model category $E$ at a class $S$ of morphisms, for checking the Quillen adjunction property of $(L \dashv R)$ it is sufficient to check that $L$ preserves cofibrations, and that $R$ takes fibrant objects $c$ of $C$ to such fibrant objects of $E$ that have the property that for all $f \in S$ the derived hom-space map $\mathbb{R}Hom(f,R(c))$ is a weak equivalence.

Cohesive $(\infty,1)$-toposes

• cohesive (∞,1)-topos

Structures in a cohesive $(\infty,1)$-topos

• . Structures in a cohesive (∞,1)-topos

Infinitesimal cohesion

• . Infinitesimal cohesion

3.) Models

We construct specific cohesive (∞,1)-toposes and discuss the nature of the general abstract structures in these models. We consider discrete cohesion, Euclidean-topological cohesion, then smooth cohesion and synthetic differential cohesion.

Discrete $\infty$-groupoids

• discrete ∞-groupoid

Euclidean-topological $\infty$-groupoids

• Euclidean-topological ∞-groupoids

Smooth $\infty$-groupoids

• smooth ∞-groupoid.

Circle $n$-bundles with connection

• circle n-bundle with connection.

$\infty$-Chern-Weil theory

• ∞-Chern-Weil theory

Synthetic differential $\infty$-groupoids

• synthetic differential ∞-groupoid

$\infty$-Lie algebroids

• ∞-Lie algebroid

$\infty$-Lie algebra cohomology

• ∞-Lie algebra cohomology

4.) Applications

We study aspects of the realization of the general abstract Chern-Weil theory in a cohesive (∞,1)-topos in the model Smooth∞Grpd.

The generalization of ordinary Chern-Weil theory in ordinary differential geometry obtained this way comes from two directions:

1. The ∞-Chern-Weil homomorphism applies to $G$-principal ∞-bundles for $G$ more general than a Lie group.

   • In the simplest case $G$ may be a higher connected cover of a Lie group, realized as a smooth n-group for $n \gt 1$. Applied to these, the $\infty$-Chern-Weil homomorphism sees fractional refinements of the ordinary differential characteristic classes as seen by the ordinary Chern-Weil homomorphism.

     This we discuss in Fractional differential characteristic classes

   • More generally, $G$ may be any smooth ∞-groupoid, for instance obtained from a general L-∞ algebra or ∞-Lie algebroid by Lie integration. In

     ∞-Chern-Simons functional

     we discuss a list of examples for which the higher parallel transport of the circle n-bundles with connection in the image of the $\infty$-Chern-Weil homomorphism reproduces action functionals of various sigma-model/Chern-Simons-like quantum field theories.

2. The $\infty$-Chern-Weil homomorphism is not just a function on cohomology sets, but an (∞,1)-functor on the full cocycle ∞-groupoids.

   The allows to access the homotopy fibers of this (∞,1)-functor. Over the trivial cocycle these encode the differential refinement of the obstruction theory associated to the underlying bare cocycle. Over nontrivial cocycles they encode the corresponding twisted cohomology.

A central class of examples are higher differential Spin structures induced from the Whitehead tower of the orthogonal group. These appear in various guises in string background gauge fields.

Higher differential spin structures

But also differential T-duality pairs are an example, as we discuss in

Differential T-duality

Finally, we observe that the ∞-Chern-Weil homomorphism may be understood as providing the Lagrangian of higher analogs of Chern-Simons theory, in that its intrinsic integration, yields a functional on the ∞-groupoid of ∞-connections that generalizes the action functional of Chern-Simons theory from ordinary semisimple Lie algebras and their Killing form to arbitrary ∞-Lie algebroids and arbitrary invariant polynomials on them. We conclude in ∞-Chern-Simons functionals by a discussion of a list of quantum field theories obtained this way.

Higher $Spin$-structures

Orientation structure

• orientation

Spin structure

• spin structure

Smooth String structure and the String-2-group

• string 2-group

Smooth Fivebrane structure and the Fivebrane 6-group

• fivebrane 6-group

Higher $Spin^c$-structures

• spin^c

• Spin^c structure

Classical supergravity

• D'Auria-Fre formulation of supergravity

• super Poincare Lie algebra

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

Twisted differential $\mathbf{c}$-structures

• twisted differential c-structures

Higher differential spin structures

• twisted spin structure

• differential string structure

  • Green-Schwarz mechanism

  • supergravity C-field

• differential fivebrane structure

Differential T-duality

• differential T-duality

Higher symplectic geometry

• symplectic Lie n-algebroid

$\infty$-Chern-Simons functionals

• ∞-Chern-Simons theory

$\infty$-Wess-Zumino-Witten functionals

• ∞-Wess-Zumino-Witten theory